Show that if f and g are arithmetic functions, F = f * g, and h is the Dirichlet inverse of g, then f = F * h.
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Suppose a tub has the shape of an elliptical paraboloid given by z = ax2 + by2 (where a, b are some positive constants). If a marble were released at the point (1, l, a + 6) on the inside surface of the tub,.

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In a finite abelian group G. If d is the order of a, and m is any number with am = e, then d divides m. Let a be an element of of order d in a finite abelian group, . Show (ii).

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The integral f(x) dx is to be approximated by the integration formula which, like the trapezoidal rule, is to be exact for all linear functions. Write down the linear polynomial p1(x) which interpolates f at x0 = 1.5 and x1 - 2. (The equation.

Show that each of the relation R in the set, given by
(i)
(ii)
is an equivalence relation. Find the set of all elements related to 1 in each case.
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Show that if d is a common divisor of x and y, then d divides ax + by for every a, b in Z. (Note: ax + by is usually referred to as an integer linear comb y simply because a and b are integers.) Let q,x,y,r in Z such.

Show briefly that 6Z is an ideal of 2Z. Find addition and multiplication table for the factor ring^2Z/6Z. Is the factor ring in part isomorphic to the field^Z_3? Why?
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The temperature at the point (x, y, z) in space is given by T(x, y, z) = x + yz. A fly is at the point (1,2,1). In what direction should he begin to fly to cool off as quickly as possible? Your answer.

Show that f f = 1A
Let A=R\{1}, and let f:A A be given by f(x) = x + 1/x - 1. Show that fdegree f = 1A
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A tea shop owner is mixing a blend of two teas, one of which costs $6.50 per pound, the other costing $4.00 per pound. The owner wants to have 20 pounds of a mixture that will sell for $5.50 per pound. How much.

show that if a polynomial k2+ak+b with integer coefficeint has a rationall root x only if x is an integer. Give an example of a polynomial that has no rational root.
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A manufacturing company receives orders for engines from two assembly plants. Plant l eeds at least 45 engines, and plant ll needs at least 32 engines. The company can send at most 120 engines to these assembly plants. It costs $30 per engine to.

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Find the standard form of the equation of the following ellipse and give the location of its foci. Find the standard form of the equation of each ellipse satisfying .he given conditions. Sketch a graph as needed. Foci:.

Show T being one-to-one...
This is the relevant text pertaining to the question;
Suppose V is finite-dimensional and let T L(V). Show that T being one-to-one is equivalent to T being onto. Feel free to introduce a basis to assist you in the proof..

Show that if P is the matrix of a projection onto a plane in R3, then rank(P) = 2.
It is Linear Algebra, please show all the steps thanks!
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Estimate a reasonable worst-case error for the approximation cos (x) = 1- x 2/2 + x 4/4! on the interval of [-pi/4,pi/4]. This should be based on the residual error of the Taylor's series when using N = 4.
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